4.2. The luminosity distance and gravitational lensing
Before proceeding to discuss possible constraints on from gravitational lensing in this section and high redshift supernovae in the next, let us introduce a quantity which plays a crucial role in these discussions, namely the luminosity distance dL(z) up to a given redshift z. Consider an object of absolute luminosity located at a coordinate distance r from an observer at r = 0. Light emitted by the object at a time t is received by the observer at t = t0, t and t0 being related by the cosmological redshift 1 + z = a(t0) / a(t). The luminosity flux reaching the observer is
where dL is the luminosity distance to the object 
The luminosity distance dL depends sensitively upon both the spatial curvature and the expansion dynamics of the universe. To demonstrate this we determine dL using the expression for the coordinate distance r obtained by setting ds2 = 0 in (2), resulting in
where = 0t cdt / a(t).
Furthermore, since dz / dt = - (1 + z)H(z), we get
where h(z) = H(z) / H0 is defined in (18), and, in a universe with several components
Substituting (27) and (26) in (23) we get the following expression for the luminosity distance in a multicomponent universe with a cosmological term 
and S(x) is defined as follows: S(x) = sin(x) if = 1 (total > 1), S(x) = sinh(x) if = - 1 (total < 1), S(x) = x if = 0 (total = 1).
Figure 4. The luminosity distance dL (in units of H0-1) is shown as a function of cosmological redshift z for flat cosmological models with a cosmological constant m + = 1. Heavier lines correspond to larger values of m. For comparison we also show (dashed line) the angular size in a flat de Sitter universe ( = 1).
Before we turn to applications, let us consider a simple example which provides us with an insight into the role played by the luminosity distance dL in cosmology. In a spatially flat universe the expression for dL simplifies considerably, so that we get for the matter dominated model (a t2/3):
On the other hand in de Sitter space (a exp(H0 t))
Comparing (30) and (31) we find dLDS(z) > dLMD(z), which means that an object located at a fixed redshift will appear brighter in an Einstein-de Sitter universe than it will in de Sitter space (equivalently in the steady state model). (5) This is also true for a two component universe consisting of matter and a cosmological constant as demonstrated in Fig 4. In a spatially flat universe the presence of a -term increases the luminosity distance to a given redshift, leading to interesting astrophysical consequences. Since the physical volume associated with a unit redshift interval increases in models with > 0, the likelihood that light from a quasar will encounter a lensing galaxy is larger in such models. Consequently the probability that a quasar is lensed by intervening galaxies increases appreciably in a dominated universe, and can be used as a test to constrain the value of [72, 71, 192]. Following [73, 26, 37] we give below the probability of a quasar at redshift zs being lensed relative to the fiducial Einstein-de Sitter model (m = 1)
where d (z1, z2) is a generalization of the angular distance dA = dL(1 + z)-2 discussed in Section 4.5:
and S(12) is defined as follows, S(12) = sin(12) if = 1 (total > 1), S(12) = sinh(12) if = -1 (total < 1), S(12) = 12 if = 0 (total = 1). In Fig 5 we show the lensing probability P(lens) for the spatially flat universe m + = 1. A large increase in the lensing probability over the fiducial m = 1 value is clearly seen in models with low m (high ). (For a broader analysis of parameter space see .)
Figure 5. The lensing probability P(lens) evaluated relative to the fiducial case m = 1 is shown as a function of for flat cosmological models m + = 1. The source redshift is taken at zs = 1, 2, 3 respectively.
Turning now to the observational situation, at the time of writing the best observational estimates give a 2 upper bound < 0.66 obtained from multiple images of lensed quasars [111, 112, 136]. Since radio sources are not plagued by some of the systematic errors arising in an optical search (notably extinction in the lens galaxy and the quasar discovery process) a search involving radio selected lenses can yield useful complementary information to optical searches . Recent work by Falco et al (1998) gives < 0.73 which is only marginally consistent with optical estimates, a combined analysis of optical and radio data yields a slightly more conservative upper bound < 0.62 at the 2 level (for flat universes) . (Constraints on from both lensing and Type 1a Supernovae are discussed in ; also see next section. An interesting new method of constraining from weak lensing in clusters is discussed in , also see  and section 4.6.) Improved understanding of statistical and systematic uncertainties combined with new surveys and better quality data promise to make gravitational lensing a powerful technique for constraining cosmological parameters and cosmological world models.
5 For instance a galaxy at redshift z = 3 will appear 9 times brighter in a flat matter dominated universe than it will in de Sitter space (see Fig 4). Back.